Shock-wave cosmology inside a black hole
Joel Smoller† and Blake Temple‡§
†Department of Mathematics, University of Michigan, Ann Arbor, MI 48109; and ‡Department of Mathematics, University of California, Davis, CA 95616
Communicated by S.-T. Yau, Harvard University, Cambridge, MA, June 23, 2003 (received for review October 17, 2002) We construct a class of global exact solutions of the Einstein the total mass M(r, t) inside radius r in the FRW metric at fixed equations that extend the Oppeheimer–Snyder model to the case time t increases like r,¶ and so at each fixed time t we must have of nonzero pressure, inside the black hole, by incorporating a shock r Ͼ 2M(r, t) for small enough r, while the reverse inequality holds wave at the leading edge of the expansion of the galaxies, for large r.Weletr ϭ rcrit denote the smallest radius at which arbitrarily far beyond the Hubble length in the Friedmann– rcrit ϭ 2M(rcrit, t).] We show that when k ϭ 0, rcrit equals the Robertson–Walker (FRW) spacetime. Here the expanding FRW Hubble length. Thus, we cannot match a critically expanding universe emerges be-hind a subluminous blast wave that explodes FRW metric to a classical TOV metric beyond one Hubble outward from the FRW center at the instant of the big bang. The length without continuing the TOV solution into a black hole, total mass behind the shock decreases as the shock wave expands, and we showed in ref. 4 that the standard TOV metric cannot be and the entropy condition implies that the shock wave must continued into a black hole. Thus to do shock matching with a weaken to the point where it settles down to an Oppenheimer– k ϭ 0 FRW metric beyond one Hubble length, we introduce the Snyder interface, (bounding a finite total mass), that eventually TOV metric inside the black hole. emerges from the white hole event horizon of an ambient Schwarzschild spacetime. The entropy condition breaks the time 2. The TOV Metric Inside the Black Hole symmetry of the Einstein equations, selecting the explosion over the implosion. These shock-wave solutions indicate a cosmological When the metric anzatz is taken to be of the TOV form model in which the big bang arises from a localized explo- sion occurring inside the black hole of an asymptotically flat ds2 ϭ ϪB͑r͒dt 2 ϩ AϪ1͑r͒dr 2 ϩ r 2d⍀2, [2.1] Schwarzschild spacetime. and the stress tensor T is taken to be that of a perfect fluid e describe a cosmological model based on matching a comoving with the metric, the Einstein equations G ϭ T, inside Wcritically expanding Friedmann–Robertson–Walker the black hole, take the form (FRW) metric to a metric that we call the Tolman– Oppenheimer–Volkoff (TOV) metric inside the black hole p ϩ ¯ NЈ Ј ϭ across a shock wave that lies beyond one Hubble length from the p Ϫ , [2.2] center of the FRW spacetime. This implies that the spacetime 2 N 1 beyond the shock wave must lie inside a black hole, thus extending the shock matching limit of one Hubble length that we N NЈ ϭ Ϫͭ ϩ prͮ, [2.3] identified in ref. 1. r In the exact solutions constructed in this article, the expanding FRW universe emerges behind a subluminous blast wave that BЈ 1 N explodes outward from the origin r ϭ 0 at the instant of the big ϭ Ϫ ͭ ϩ ͮ. [2.4] B N Ϫ 1 r bang t ϭ 0, at a distance beyond one Hubble length.¶ The shock wave then continues to weaken as it expands outward until the Hubble length eventually catches up to the shock, and this marks We let , p denote the density and pressure, respectively, and r the event horizon of a black hole in the TOV metric beyond the is taken to be the timelike variable because we assume shock. From this time onward, the shock wave is approximated by a zero pressure (k ϭ 0) Oppenheimer–Snyder (OS) interface, 2M͑r͒ A͑r͒ ϭ 1 Ϫ ϵ 1 Ϫ N͑r͒ Ͻ 0. [2.5] and thus the OS solution gives the large time asymptotics of these r solutions. Surprisingly, the equation of state p ϭ 1 of early big 3 bang physics is distinguished by the differential equations, and Here, M(r) has the interpretation as the total mass inside the ball only for this equation of state does the shock wave emerge from of radius r when r Ͼ 2M, but M does not have the same the big bang at a finite nonzero speed, the speed of light. This 1 interpretation inside the black hole because r Ͻ 2M. The system is surprising because the equation of state p ϭ played no 3 2.2–2.4 defines the simplest class of gravitational metrics that special role in shock matching outside the black hole (2). We find it interesting that such a shock wave emerging from the big bang contain matter and evolve inside the black hole. beyond the Hubble length would account for the thermalization
of radiation in a region well beyond the light cone of an observer Abbreviations: FRW, Friedmann–Robertson–Walker; TOV, Tolman–Oppenheimer–Volkoff; positioned at the FRW center at present time, even though the OS, Oppenheimer–Snyder. FRW expansion is finite, and the model does not invoke §To whom correspondence should be addressed. E-mail: [email protected]. inflation. Details will appear in our forthcoming article (3); here, ¶We let (t, r) denote standard FRW coordinates, so that r ϭ rR(t) measures arclength we summarize this work and describe its physical interpretation. distance at each fixed value of the FRW time t, where R denotes the cosmological scale factor. Barred coordinates also refer to TOV standard coordinates, in which case r ϭ rR(t) 1. Statement of the Problem also holds as a consequence of shock matching. If there is a shock wave at the leading edge of the expansion of System 2.2–2.4 for A Ͻ 0 differs substantially from the TOV equations for A Ͼ 0 because, for example, the energy density T00 is equated with the timelike component Grr when A Ͻ the galaxies, then we can ask what is the critical radius rcrit at each 0, but with Gtt when A Ͼ 0. In particular, this implies that MЈϭϪ4r2 when A Ͻ 0, versus fixed time t in a k ϭ 0 FRW metric such that the total mass inside MЈϭ4r2 when A Ͼ 0, the latter being the equation that gives the mass function its a shock wave positioned beyond that radius puts the universe physical interpretation outside the black hole. inside a black hole? [There must be such a critical radius because © 2003 by The National Academy of Sciences of the USA
11216–11218 ͉ PNAS ͉ September 30, 2003 ͉ vol. 100 ͉ no. 20 www.pnas.org͞cgi͞doi͞10.1073͞pnas.1833875100 Downloaded by guest on September 25, 2021 3. Shock Matching Inside the Black Hole Eq. 3.1 uncouples from 3.2, and thus solutions of system 3.1–3.3 Matching a given k ϭ 0 FRW metric to a TOV metric inside the are determined by the scalar nonautonomous Eq. 3.1. Making ϭ ͞ black hole across a shock interface, leads to the system of the change of variable S 1 N, which transforms the big bang 3 ϱ 3 ordinary differential equations N over to rest point at S 0, cf. ref. 1, we obtain, du ͑1 ϩ u͒ ͑3u Ϫ 1͒͑ Ϫ u͒ ϩ 6u͑1 ϩ u͒S du ͑1 ϩ u͒ ͑3u Ϫ 1͒͑ Ϫ u͒N ϩ 6u͑1 ϩ u͒ ϭ ͭ ͮͭ ͮ ϭ Ϫͭ ͮͭ ͮ, ͑ ϩ ͒ ͑ Ϫ ͒ ϩ ͑ ϩ ͒ . dN 2͑1 ϩ 3u͒N ͑ Ϫ u͒N ϩ ͑1 ϩ u͒ dS 2 1 3u S u 1 u S [3.1] [4.4] dr 1 r We take as entropy condition 3.6, and in addition, we require ϭ Ϫ , [3.2] that the TOV equation of state meet the physical bound dN 1 ϩ 3u N 0 Ͻ p Ͻ . [4.5] with conservation constraint The conditions N Ͼ 1 and 0 Ͻ p Ͻ p restrict the domain of 4.4 to Ϫ͑1 ϩ u͒ ϩ ͑ Ϫ u͒N Ͻ Ͻ Ͻ Ͻ Ͻ ϭ . [3.3] the region 0 u 1, 0 S 1. Inequalities 3.6 and 4.5 ͑1 ϩ u͒ ϩ ͑ Ϫ u͒N are both implied by the single condition Here 1 Ϫ u Ϫ u S Ͻ ͩ ͪͩ ͪ. [4.6] 1 ϩ u ϩ u p p u ϭ , v ϭ , ϭ , [3.4] We prove the following theorem: ϭ ͞ and p denote the (known) FRW density and pressure, p , Theorem 1. For every ,0Ͻ Ͻ 1, there exists a unique solution and all variables are evaluated at the shock. Solutions of 3.1–3.3 u(S) of 4.4, such that 4.6 holds on the solution for all S,0Ͻ S Ͻ determine the (unknown) TOV metrics that match the given 1, and on this solution,0Ͻ u(S) Ͻ u, limS30 u(S) ϭ u, where
FRW metric Lipschitz continuously across a shock interface, u ϭ Min{1͞3, }, and APPLIED
such that conservation of energy and momemtum hold across the MATHEMATICS shock, and such that there are no delta function sources at the lim p ϭ 0 ϭ lim p. [4.7] shock (4, 5). S31 S31 For such solutions, the speed of the shock interface relative to Concerning the the shock speed, we have: the fluid comoving on the FRW side of the shock, is given by Ͻ Ͻ Ϫ u Theorem 2. Let 0 1. Then the shock wave is everywhere s ϭ Rr˙ ϭ ͱN ͩ ͪ. [3.5] subluminous, that is, the shock speed s(S) ϵ s(u(S)) Ͻ 1 for all 1 ϩ u 0 Ͻ S Յ 1, if and only if Յ 1͞3. ASTRONOMY By a careful analysis of the asymptotics of the solution near Note that the dependence of 3.1–3.3 on the FRW metric is only S ϭ 0, we can prove through the variable . Since solutions of 3.1–3.3 are formally time-reversible but shock waves are not, system 3.1–3.3 must be Theorem 3. The shock speed at the big bang S ϭ 0 is given by: augmented by an entropy condition for shocks that breaks the time symmetry. Since we are interested in solutions that model lim s͑S͒ ϭ 0, Ͻ 1͞3, [4.8] the ‘‘big bang’’ as a localized explosion with an outgoing blast S30 wave emanating from r ϭ 0 at time t ϭ 0, we impose the entropy ͑ ͒ ϭ ϱ Ͼ ͞ conditions, lim s S , 1 3, [4.9] S30 0 Ͻ p Ͻ p,0Ͻ Ͻ . [3.6] lim s͑S͒ ϭ 1, ϭ 1͞3. [4.10] 3 Condition 3.6 for outgoing shock waves implies that the shock S 0 wave is compressive and is sufficient to rule out unstable 5. Bounds on the Shock Position rarefaction shocks in classical gas dynamics. be the FRW radial position of the shock wave at the instant ءLet r ϭ ϭ R(0) 0 whenءr ءExact Shock-Wave Solutions Inside the Black Hole of the big bang [the arclength distance r .4 R(0) ϭ 0]. The analysis implies that the shock wave will first In the case when the FRW pressure is given by the equation of become visible at the FRW center r ϭ 0 at the moment t ϭ t0 state ϭ Ϫ1 ϭ Ϫ1 [R(t0) 1], when the Hubble length H0 H (t0) satisfies ϭ p , [4.1] 1 1 ϩ 3 [5.1] ,ءϭ r assumed constant, 0 Ͻ Ͻ 1, the FRW equations have the H0 2 exact solutions is the FRW position of the shock at the instant of the ءwhere r 4 1 big bang. At this time, we prove that the number of Hubble ϭ , [4.2] lengths ͌N from the FRW center to the shock wave at time t ϭ 3͑1 ϩ ͒2 t2 0 t0 can be estimated by 2 3͑1ϩ͒ 1ϩ3 t 2 2 ͱ ͩ ͪ ϭ ͩ ͪ Յ Յ ͱ Յ 3 ϩ R , [4.3] 1 N0 e 1 . t0 1 ϩ 3 1 ϩ 3 which assumes an expanding universe, (R˙ Ͼ 0), and initial Thus, in particular, the shock wave will still lie beyond the conditions R ϭ 0att ϭ 0, and R ϭ 1, at t ϭ t0. Since is constant, Hubble length 1͞H0 at the FRW time t0 when it first becomes
Smoller and Temple PNAS ͉ September 30, 2003 ͉ vol. 100 ͉ no. 20 ͉ 11217 Downloaded by guest on September 25, 2021 visible. Furthermore, we prove that the time tcrit Ͼ t0 at which the exactly one Hubble length from the FRW center r ϭ 0, and then shock wave will emerge from the white hole given that t0 is the continues on out to infinity along a geodesic of the Schwarzschild first instant at which the shock becomes visible at the FRW metric outside the black hole. These solutions thus indicate a center, can be estimated by scenario for the big bang in which the expanding universe emerges from an explosion emanating from a white hole singu- 1 2ͱ3 2 tcrit 2 larity inside the event horizon of an asymptotically flat Schwar- Յ Յ ϩ ϩ e4 ϩ e 1 , [5.2] 1 3 t0 1 3 zschild spacetime of finite mass. The model does not require the physically implausible assumption that the uniformly expanding for 0 Ͻ Յ 1͞3, and by the better estimate portion of the universe is of infinite mass and extent at every ͱ fixed time, and it has the nice feature that it embeds the big bang 6 tcrit 3 singularity of cosmology within the event horizon of a larger e 4 Յ Յ e2, [5.3] t0 spacetime, the Schwarzschild spacetime. Moreover, the model also allows for the uniform expansion of arbitrarily large den- in the case ϭ 1͞3. For example, 5.2 and 5.3 imply that at the sities within an arbitrarily large mass extended over an arbitrary OS limit ϭ 0, number of Hubble lengths early on in the big bang, a prerequisite for the standard physics of the big bang at early times. t ͱ ϭ crit ϭ We conclude that these shock-wave solutions give the global N0 2, 2, t0 dynamics of strong gravitational fields in an exact solution, the dynamics is qualitatively different from the dynamics of solutions ϭ ͞ and in the limit 1 3, when the pressure p ϵ 0, the solution suggests a big bang cosmological model in which the expanding universe is bounded t Յ crit Յ Ͻ ͱ Յ throughout its expansion, and the equation of state most relevant 1.8 4.5, 1 N0 4.5. 1 t0 at the big bang, p ϭ , is distinguished by the differential 3 equations. But these solutions are only rough qualitative models We conclude in these shock-wave cosmological models that because the equation of state on the TOV side is determined by the moment t0 when the shock wave first becomes visible at the the equations and therefore cannot be imposed. That is, the TOV FRW center it must lie within 4.5 Hubble lengths of the center. density and pressure p only satisfy the entropy conditions (3.6), Throughout the expansion up until this time, the expanding together with the loose physical bounds (4.5); and on the FRW universe must lie entirely within a black hole; the universe will side, the equation of state is taken to be p ϭ , ϵ constant, eventually emerge from this black hole, but not until some later 0 Ͻ Ͻ 1. Nevertheless, these bounds imply that the equations time tcrit, where tcrit does not exceed 4.5 t0. of state are qualitatively reasonable, and we expect that these solutions will capture the gross dynamics of solutions when more 6. Concluding Remarks general equations of state are imposed. For more general We have constructed global exact solutions of the Einstein equa- equations of state, other waves, (e.g. rarefaction waves), would tions in which the expanding FRW universe extends out to a shock need to be present to meet the conservation constraint and wave that lies arbitrarily far beyond the Hubble length. The distance thereby mediate the transition across the shock wave. Such to the shock wave at any given value of the Hubble constant is transitional waves would be pretty much impossible to model in determined by one free parameter that can be taken to be the FRW an exact solution, but the fact that we can find global solutions position of the shock wave at the instant of the big bang. that meet our physical bounds, and that are qualitatively the The critical OS solution inside the black hole is obtained in the same, for all values of ʦ (0, 1), and all initial shock positions, limit of zero pressure and provides the large time asymptotics, leads us to expect that such a shock wave would be the dominant but the shock-wave solutions differ qualitatively from the OS wave in a large class of problems. solution. For example, the shock wave models contain matter, As a final remark, we note that because Einstein’s theory by and thus do not rely on any portion of the empty space itself does not choose an orientation for time, it follows that if Schwarzschild metric inside the black hole for their construction. we believe that a black hole can exist in the forward time collapse In both models, the interface propagates outward from the FRW of a mass through an event horizon as t 3 ϱ, (the time t as center r ϭ 0 at the instant of the big bang, but in the shock-wave observed in the far field), then we must also allow for the model the mass function M(r, t) is infinite at the instant of the possibility of the time reversal of this process, a white hole big bang and immediately becomes a finite decreasing function explosion of matter out through an event horizon coming from of FRW time, for all future times t Ͼ 0. And although the OS t 3 Ϫϱ. That is, if we are willing to accept black holes in the solution is time reversible, the directional orientation of the forward time dynamics of astrophysical objects whose collapse shock interface relative to the various observers is determined by appears so great as to form an event horizon in the future, then an entropy condition (6–8). The entropy condition selects the by symmetry, we may well also be forced to accept white holes explosion over the implosion, and the condition that the entropy in the backward time dynamics of astrophysical objects, which, condition be satisfied globally, determines a unique solution. like the expanding universe, appear to have expansions so great Since the TOV radial coordinate r is timelike inside the black as to have emerged from an event horizon in the past. Given this, hole, we can also say that the density (r) and mass M(r) are both it is natural to wonder if there is a connection between the mass constant at each fixed time in the TOV spacetime beyond the that disappears into black hole singularities and the mass that shock wave. emerges from white hole singularities. The shock interface that marks the boundary of the FRW expansion continues out through the white hole event horizon of This work was supported in part by National Science Foundation Applied an ambient Schwarzschild metric at the instant when the wave is Mathematics Grants DMS-010-3998 (to J.S.) and DMS-010-2493 (to B.T.).
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